cess for a magnetic bottle

In this appendix we discuss some details of the transformation of a
Hamiltonian with a quadratic part (34)
into generalized normal form. More
specifically, we show how to simplify the Lie operator

which is adjoint to this particular .

The unitary matrix that by a similarity transformation puts the
Hamiltonian matrix

into the Jordan normal form of (35) is

with being the canonical base vectors of . This formula can easily be derived by performing a certain permutation of the rows and columns first, followed by a transformation similar to (15).

In the new coordinates
the Lie operator takes
on the form

This representation of the Lie operator is advantageous, because here we have collected as many non-zero entries (of the matrix representation) of on the diagonal as possible. Only the first sum yields an off-diagonal contribution.

We have not yet
made any assumptions about the ordering of the
monomials
in the basis of .
If one
chooses the *lexicographical ordering* [2] of the
basis monomials,
then the
matrix representation of
becomes an upper diagonal matrix
for all , and all the manipulations of that are necessary in the
course of the normalization procedure (solving linear equations, inverting
, ...) become easier and consume much less computing time.

In the case it is possible to achieve even further simplification
by an appropriate ordering of the basis monomials of .
One can introduce the so-called *magnetic bottle ordering* of
monomials which results in
being bidiagonal.